Does a smooth mapping always have an inverse map which is also smooth?

Question

If not, can someone provide counterexamples? Thank you

Answer 1

First of all, not every smooth map is invertible as it may not be a bijection. Secondly, even if it is invertible, its inverse may not be smooth. Consider the map $f : \mathbb{R} \to \mathbb{R}$, $f(x) = x^3$. This is a smooth map which is also a bijection, so it has an inverse, namely $f^{-1}(x) = \sqrt[3]{x}$. Note that $f^{-1}$ is not differentiable at $0$ so it certainly isn't smooth.

Answer 2

$f(x) = x^3$ is a counterexample.